Contents The Stochastic Growth Model Contents Introduction The stochastic growth model The steady state Linearization around the balanced growth path Solution of the linearized model Impulse response functions 13 Conclusions 18 Appendix A A1 The maximization problem of the representative firm A2 The maximization problem of the representative household 20 20 20 Appendix B 22 Appendix C C1 The linearized production function C2 The linearized law of motion of the capital stock C3 The linearized first-order condotion for the firm’s labor demand C4 The linearized first-order condotion for the firm’s capital demand C5 The linearized Euler equation of the representative household C6 The linearized equillibrium condition in the goods market 24 24 25 26 26 28 30 References 32 www.sylvania.com We not reinvent the wheel we reinvent light Fascinating lighting offers an ininite spectrum of possibilities: Innovative technologies and new markets provide both opportunities and challenges An environment in which your expertise is in high demand Enjoy the supportive working atmosphere within our global group and beneit from international career paths Implement sustainable ideas in close cooperation with other specialists and contribute to inluencing our future Come and join us in reinventing light every day Light is OSRAM Download free eBooks at bookboon.com Click on the ad to read more The Stochastic Growth Model Introduction Introduction This article presents the stochastic growth model The stochastic growth model is a stochastic version of the neoclassical growth model with microfoundations,1 and provides the backbone of a lot of macroeconomic models that are used in modern macroeconomic research The most popular way to solve the stochastic growth model, is to linearize the model around a steady state,2 and to solve the linearized model with the method of undetermined coefficients This solution method is due to Campbell (1994) The set-up of the stochastic growth model is given in the next section Section solves for the steady state, around which the model is linearized in section The linearized model is then solved in section Section shows how the economy responds to stochastic shocks Some concluding remarks are given in section Download free eBooks at bookboon.com The stochastic growth model The Stochastic Growth Model The stochastic growth model The representative firm Assume that the production side of the economy is represented by a representative firm, which produces output according to a Cobb-Douglas production function: Yt = Ktα (At Lt )1−α with < α < (1) Y is aggregate output, K is the aggregate capital stock, L is aggregate labor supply and A is a technology parameter The subscript t denotes the time period The aggregate capital stock depends on aggregate investment I and the depreciation rate δ: Kt+1 = (1 − δ)Kt + It with ≤ δ ≤ (2) The productivity parameter A follows a stochastic path with trend growth g and an AR(1) stochastic component: ln At = ln A∗t + Aˆt Aˆt = φA Aˆt−1 + εA,t A∗t = A∗t−1 (1 with |φA | < (3) + g) The stochastic shock εA,t is i.i.d with mean zero The goods market always clears, such that the firm always sells its total production Taking current and future factor prices as given, the firm hires labor and invests in its capital stock to maximize its current value This leads to the following first-order-conditions:3 (1 − α) Yt Lt = wt = Et (4) 1−δ Yt+1 α + Et + rt+1 Kt+1 + rt+1 (5) According to equation (4), the firm hires labor until the marginal product of labor is equal to its marginal cost (which is the real wage w) Equation (5) shows that the firm’s investment demand at time t is such that the marginal cost of investment, 1, is equal to the expected discounted marginal product of capital at time t + plus the expected discounted value of the extra capital stock which is left after depreciation at time t + Download free eBooks at bookboon.com The stochastic growth model The Stochastic Growth Model The government The government consumes every period t an amount Gt , which follows a stochastic path with trend growth g and an AR(1) stochastic component: ˆt ln Gt = ln G∗t + G ˆ t = φG G ˆ t−1 + εG,t G with |φG | < (6) G∗t = G∗t−1 (1 + g) The stochastic shock εG,t is i.i.d with mean zero εA and εG are uncorrelated at all leads and lags The government finances its consumption by issuing public debt, subject to a transversality condition,4 and by raising lump-sum taxes.5 The timing of taxation is irrelevant because of Ricardian Equivalence.6 360° thinking Discover the truth at www.deloitte.ca/careers Download free eBooks at bookboon.com © Deloitte & Touche LLP and affiliated entities Click on the ad to read more The stochastic growth model The Stochastic Growth Model The representative household There is one representative household, who derives utility from her current and future consumption: ∞ Ut = Et s=t 1+ρ s−t ln Cs with ρ > (7) The parameter ρ is called the subjective discount rate Every period s, the household starts off with her assets Xs and receives interest payments Xs rs She also supplies L units of labor to the representative firm, and therefore receives labor income ws L Tax payments are lump-sum and amount to Ts She then decides how much she consumes, and how much assets she will hold in her portfolio until period s + This leads to her dynamic budget constraint: Xs+1 = Xs (1 + rs ) + ws L − Ts − Cs (8) We need to make sure that the household does not incur ever increasing debts, which she will never be able to pay back anymore Under plausible assumptions, this implies that over an infinitely long horizon the present discounted value of the household’s assets must be zero: s lim Et s→∞ 1 + rs ′ s′ =t Xs+1 = (9) This equation is called the transversality condition The household then takes Xt and the current and expected values of r, w, and T as given, and chooses her consumption path to maximize her utility (7) subject to her dynamic budget constraint (8) and the transversality condition (9) This leads to the following Euler equation:7 Cs = Es + rs+1 1 + ρ Cs+1 (10) Equilibrium Every period, the factor markets and the goods market clear For the labor market, we already implicitly assumed this by using the same notation (L) for the representative household’s labor supply and the representative firm’s labor demand Equilibrium in the goods market requires that Yt = Ct + It + Gt Equilibrium in the capital market follows then from Walras’ law Download free eBooks at bookboon.com (11) ... Click on the ad to read more The Stochastic Growth Model Introduction Introduction This article presents the stochastic growth model The stochastic growth model is a stochastic version of the neoclassical...Contents The Stochastic Growth Model Contents Introduction The stochastic growth model The steady state Linearization around the balanced growth path Solution of the linearized model Impulse... how the economy responds to stochastic shocks Some concluding remarks are given in section Download free eBooks at bookboon.com The stochastic growth model The Stochastic Growth Model The stochastic